dynamical systems analysis

Existing Koopman-based methods struggle to simultaneously learn continuous-time dynamics and guarantee stability for unknown nonlinear systems under low-frequency sampling. Method: This paper proposes a novel framework that jointly learns the Koopman generator and a Lyapunov function, tightly integrating high-accuracy Koopman generator estimation with physics-informed neural networks (PINNs) and embedding Lyapunov stability constraints. Attractor domain verification is performed formally using an SMT solver. Contribution/Results: The approach significantly expands the verifiable region of the estimated region of attraction while reducing conservatism. It provides mathematically certified stability guarantees for dynamical systems under sparse observations—overcoming a key limitation of current Koopman methods, which lack rigorous stability certification under low-sampling-rate conditions.

Traditional linearization-based analyses often fail to accurately characterize the stability of optimization algorithms under nonlinear dynamics, leading to potential misjudgments. This work addresses this limitation by explicitly analyzing the nonlinear behavior of gradient descent (GD) and stochastic gradient descent (SGD) near minima. It proposes a multivariate GD stability criterion based on higher-order derivatives that captures stable oscillatory regimes beyond the reach of linear analysis. Furthermore, the study reveals that the overall stability of SGD can be dominated by a single unstable batch rather than governed by averaging effects. Theoretically, it is proven that when all batches are linearly stable, SGD is nonlinearly stable in expectation; however, the presence of even one unstable batch is sufficient to induce global divergence.

Deep learning training often suffers sudden collapses due to minute perturbations, undermining reproducibility and scalability. This work reframes training stability as an intrinsic property of the learning system through the lens of dynamical systems theory, proposing a unified analytical framework that integrates optimization dynamics, data structure, parameter evolution, and learning signals. The authors introduce a controlled perturbation auditing method to quantify how training trajectories respond to structured disturbances. Their analysis reveals three key principles: high performance and stability are frequently decoupled; controlled randomness generally enhances robustness; and low-dimensional latent meta-state deviations consistently precede performance collapse. These findings are validated across both reinforcement learning and large language models, offering a measurable, comparable, and actionable theoretical foundation for understanding learning dynamics beyond final performance metrics.

Traditional stability and sensitivity analyses rely on known governing equations and linearization assumptions, rendering them inadequate for nonlinear or model-unknown complex systems. This work proposes a purely data-driven framework that leverages a neural network-based dynamic simulator combined with automatic differentiation to directly extract the system’s Jacob日晚间 matrix from observational data, thereby computing eigenmodes and resolvent modes without any prior knowledge of the governing equations. The method enables, for the first time, fully automated identification of stability properties and optimal forcing responses in nonlinear systems, transcending the limitations of classical linear theory. Experiments on chaotic systems and high-dimensional fluid flows demonstrate that the framework accurately captures dominant instability modes and input–output structures even in strongly nonlinear regimes.

Integrating physics-based and machine learning models for dynamic system modeling remains challenging due to difficulties in unifying multi-model representations, handling algebraic loops, and managing discontinuous events within a coherent framework. Method: This paper proposes a learnable and interpretable hybrid modeling paradigm built upon a novel wildcard architecture—the first to enable unified symbolic representation of algebraic, discrete, and differential equations—supporting end-to-end differentiable joint optimization of physics-informed and data-driven components. Grounded in systems theory and symbolic modeling principles, the approach inherently avoids algebraic loops and explicitly models discontinuities. Contribution/Results: Experiments demonstrate that the framework automatically identifies and resolves diverse dual-model compositions, achieving significant improvements over state-of-the-art methods in prediction accuracy, model interpretability, and cross-scenario generalizability.

This work addresses the challenge of convergence failure in inverse parallel solvers for nonlinear systems of equations, which often arises due to oscillatory or chaotic dynamics. To enhance stability, the authors propose an adaptive stabilization mechanism based on the local maximum Lyapunov exponent (LLE). By estimating the LLE via k-nearest neighbors and integrating it with sliding-window micro-time-series analysis, the method enables real-time detection of unstable phases along the solution trajectory. A Lyapunov-guided parameter control strategy is then developed to dynamically adjust solver parameters, thereby reinforcing numerical stability. Experimental results demonstrate strong agreement between theoretical stability diagrams and empirical Lyapunov profiles, confirming that the proposed approach significantly improves the robustness and convergence performance of solvers under perturbed initial conditions.

This work addresses the problem of classifying trajectories generated by distinct nonlinear dynamical systems, where each class corresponds to a unique system. The authors propose Dynafit, a novel method that, for the first time, integrates the Koopman operator framework with kernel methods to achieve global linearization of dynamics in a reproducing kernel Hilbert space. By leveraging the kernel trick, Dynafit efficiently computes dynamical distances between trajectories while allowing incorporation of prior knowledge. The approach demonstrates significant performance gains over baseline methods across three diverse tasks: detecting chaos in logistic maps, recognizing handwritten dynamics, and classifying visual dynamic textures. These results validate Dynafit’s effectiveness and generality in multi-class classification of nonlinear dynamical systems.

This study addresses the challenge of global linear modeling and control for highly nonlinear dynamical systems by leveraging Koopman operator theory. By introducing observable functions, the nonlinear dynamics are lifted into a higher-dimensional space where they admit an approximately linear representation. A data-driven surrogate model is constructed through a synergistic integration of Extended Dynamic Mode Decomposition (EDMD), kernelized EDMD, and machine learning techniques. The work innovatively extends the Koopman framework to input-affine systems, proposing a unified modeling approach and a corresponding Koopman-based Model Predictive Control (MPC) design methodology. Numerical simulations demonstrate that the proposed method achieves high-fidelity modeling accuracy and effective closed-loop control performance. Full reproducibility is supported by the accompanying open-source implementation.

This work addresses the lack of theoretical guarantees for the reliability of Koopman eigenpairs computed from noisy data in data-driven spectral analysis. It introduces, for the first time, shadowing trajectory theory combined with backward error analysis to interpret the residual of eigenpairs obtained via Extended Dynamic Mode Decomposition (EDMD) as an operator perturbation of the original dynamical system. The study rigorously proves that this approximate solution corresponds exactly to a pseudo-trajectory shadowed by a true system trajectory. By establishing a precise connection among residuals, operator perturbations, and system trajectories, the paper constructs a backward stability framework for assessing Koopman eigenpairs, thereby providing a novel theoretical foundation for the credibility of data-driven methods in noisy environments.

High-dimensional spatiotemporal chaotic systems are often dominated by continuous spectra, yet existing data-driven approaches frequently suffer from instability, limited interpretability, and poor scalability. This work proposes KoopGen—a generator-based neural Koopman framework that explicitly decomposes dynamics into conservative (skew-adjoint) and dissipative (self-adjoint) components via a state-dependent Koopman generator, while rigorously embedding operator-theoretic constraints. Notably, KoopGen achieves the first explicit separation of self-adjoint and skew-adjoint parts within the generator without relying on finite-dimensional assumptions or explicit spectral parameterizations. Experiments ranging from nonlinear oscillators to high-dimensional chaotic systems demonstrate that KoopGen substantially improves long-term prediction accuracy and stability, uncovering learnable and interpretable structural components underlying continuous-spectrum dynamics.